Can we say that $\mu_x$ is a prime polynomial in $\Bbb K[X]$?

Question

Let $\Bbb K$ be a field. Let $A$ be an algebra over $\Bbb K.$ Let $x \in A$ be algebraic over $\Bbb K.$ Let $\mu_x$ be the minimal polynomial of $x$ over $\Bbb K.$ Can we say that $\mu_x$ is irreducible in $\Bbb K[X]$?

What I can see is that if $A$ is an integral domain then $\mu_x$ is indeed irreducible in $\Bbb K[X].$ I don't think that this assertion holds even if $A$ is not an integral domain. Am I correct in my argument?

Any valuable suggestion regarding this will be highly appreciated. Thank you very much.

Answer 1

I got an example. $\Bbb R^2$ is not an integral domain but it is an algebra over $\Bbb R.$ Take $x=(1,0) \in \Bbb R^2.$ Then $\mu_x = X(X-1) \in \Bbb R[X]$ which is not irreducible in $\Bbb R[X].$